Alternating Sequence

Date: 01/27/97 at 20:43:09
From: KEITH MCKENZIE
Subject: I need help

Dear Dr. Math,

I need help. I have a problem that neither my friends nor I can get.  
I thought you could help.  We have to find a pattern and find the next 
three numbers for this: 0, 8, 27, _, _, _.  I hope you can find the 
answer.
			
Sincerely,
Shoushou

Date: 01/29/97 at 16:53:47
From: Doctor Wallace
Subject: Re: I need help

Dear Shoushou,

For this series, the first thing to notice is that all of the numbers 
are perfect cubes:  0 = 0^3, 8 = 2^3, and 27 = 3^3.

If we list the numbers that are the cube roots of these numbers, we 
get this series:

           0, 2, 3...

Now, all you have to do is find a pattern among these three.  The one 
I see is an alternating series.  There may be others.  Let me know how 
your work goes on this problem.

-Doctor Wallace,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 01/29/97 at 18:49:05
From: KEITH MCKENZIE
Subject: Re: I need help

This is Shoushou.  Thanks for giving me that great hint.   I'm not 
sure if I have it right, but I think it might be +2, +1, +2, +1.  This 
doesn't quite satisfy me, but it's the best I can come up with for 
now.

Sincerely,
Shoushou

Date: 01/30/97 at 12:37:59
From: Doctor Wallace
Subject: Re: I need help

Dear Shoushou,

I think you're right.  That's what I came up with, too.  And I feel 
the same way - I wasn't quite satisfied.  Somehow, it didn't feel 
just right.  I think it was because there were only 3 numbers given, 
and it's hard to come to just one pattern with only three numbers. But 
it seems that, since the number 1 was missing, +2 +1 +2 +1 ... is the 
only thing that makes sense.

Then the next two numbers in the pattern would be 5^3 and 6^3.  What 
did your teacher have in mind for the answer?  If you can come up with 
another pattern that fits the series, please write and let me know.

And write again any time you have other math questions!

-Doctor Wallace,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/